You have found the following ages (in years) of all 4 snakes at your local zoo: $ 11,\enspace 3,\enspace 5,\enspace 6$ What is the average age of the snakes at your zoo? What is the variance? You may round your answers to the nearest tenth.
Solution: Because we have data for all 4 snakes at the zoo, we are able to calculate the population mean $({\mu})$ and population variance $({\sigma^2})$ To find the population mean , add up the values of all $4$ ages and divide by $4$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{4}} x_i}{{4}} $ $ {\mu} = \dfrac{11 + 3 + 5 + 6}{{4}} = {6.3\text{ years old}} $ Find the squared deviations from the mean for each snake. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $11$ years $4.7$ years $22.09$ years $^2$ $3$ years $-3.3$ years $10.89$ years $^2$ $5$ years $-1.3$ years $1.69$ years $^2$ $6$ years $-0.3$ years $0.09$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{22.09} + {10.89} + {1.69} + {0.09}} {{4}} $ $ {\sigma^2} = \dfrac{{34.76}}{{4}} = {8.69\text{ years}^2} $ The average snake at the zoo is 6.3 years old. The population variance is 8.69 years $^2$.